Calculate the global warming time. This is the time interval from the original steady state (Problem 9.6) to the new steady state (Problem 9.7), during which the average temperature of Earth increases...

Calculate the global warming time. This is the time interval from the original steady state (Problem 9.6) to the new steady state (Problem 9.7), during which the average temperature of Earth increases by ΔT = 0.28 K. During the same time interval, the Earth area-averaged excess solar heat flux that is absorbed by the Earth surface is q′′ = 0.85 W∕m2 (Problem 9.7). Assume that the thermal inertia of the “Earth surface” is dominated by the thermal inertia of the oceans and the ocean depth averaged over the entire Earth is on the order of 1 km.

Problem 9.7

Read the preceding problem statement and consider the impact of a sudden change in the radiative properties of the atmosphere. The albedo increases from 𝜌₁
= 0.3 to 𝜌₂
= 0.3 + 𝛽, where 𝛽 ≪ 0.3, and the greenhouse factor increases from 𝛾₁
= 0.4 to 𝛾₂
= 0.4 + 𝜀, where 𝜀 ≪ 0.4. The Earth surface temperature does not have time to change: It is still at the level T₁
= 288.7 K calculated in the preceding problem by using 𝜌₁
and 𝛾₁. This creates an imbalance between the solar heat input q_s2
(associated with 𝜌₂) and the rejected heat current q_∞2
= 4𝜋R²(1 − 𝛾2)𝛼T⁴
₁
. Measurements show that the heat input imbalance is (q_s2
− q_∞2)∕(4𝜋R²) = 0.85 W∕m². Find what relation must exist between 𝛽 and 𝜀 such that this heat current imbalance occurs. If enough time passes, the Earth surface temperature rises to a new steady-state level T²
such that the rejected heat current matches the solar input q_s2. Show that the final global warming effect is T₂
− T₁
≅ 0.3 K.